Let $z$ be a nonreal complex number such that $|z| = 1.$  Find the real part of $\frac{1}{1 - z}.$
Let $z = x + yi,$ where $x$ and $y$ are real numbers.  Then from the condition $|z| = 1,$ $\sqrt{x^2 + y^2} = 1,$ so $x^2 + y^2 = 1.$

Now,
\begin{align*}
\frac{1}{1 - z} &= \frac{1}{1 - x - yi} \\
&= \frac{1 - x + yi}{(1 - x - yi)(1 - x + yi)} \\
&= \frac{1 - x + yi}{(1 - x)^2 + y^2} \\
&= \frac{1 - x + yi}{1 - 2x + x^2 + y^2} \\
&= \frac{1 - x + yi}{2 - 2x}.
\end{align*}The real part of this complex number is $\frac{1 - x}{2 - 2x} = \frac{1 - x}{2(1 - x)} = \boxed{\frac{1}{2}}.$